<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 13 Issue 3" %>
Periodicity of Non-Central Integral Arrangements Modulo Positive Integers
Aaron D. Jaggard1 and Joseph J2. Marincel3
1Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
2Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
3Department of Mathematics, Hokkaido University, North 10, West 8, Kita-ku, Sapporo, 060-0810, Japan
Annals of Combinatorics 15 (3) pp.449-464 July, 2011
AMS Subject Classification: 32S22; 52C35
An integral coefficient matrix determines an integral arrangement of hyperplanes in Rm. After modulo q reduction (q ∈ Z>0), the same matrix determines an rrangement
Aq of “hyperplanes” in Zmq . In the special case of central arrangements, Kamiya, Takemura, and Terao [J. Algebraic Combin. 27(3), 317–330 (2008)] showed that the cardinality of the complement of Aq in Zmq is a quasi-polynomial in q ∈ Z>0. Moreover, they proved in the central case that the intersection lattice of Aq is periodic from some q on. The present paper generalizes these results to the case of non-central arrangements. The paper also studies the arrangement ˆB [0,a]
m of Athanasiadis [J. Algebraic Combin. 10(3), 207–225 (1999)] to illustrate our results.
Keywords: characteristic quasi-polynomial, elementary divisor, hyperplane arrangement, intersection poset


1. Athanasiadis, C.A.: Extended Linial hyperplane arrangements for root systems and a conjecture of Postnikov and Stanley. J. Algebraic Combin. 10(3), 207–225 (1999)

2. Kamiya, H., Orlik, P., Takemura, A., Terao, H.: Arrangements and ranking patterns. Ann. Combin. 10(2), 219–235 (2006)

3. Kamiya, H., Takemura, A., Terao, H.: Periodicity of hyperplane arrangements with integral coefficients modulo positive integers. J. Algebraic Combin. 27(3), 317–330 (2008)

4. Kamiya, H., Takemura, A., Terao, H.: The characteristic quasi-polynomials of the arrangements of root systems and mid-hyperplane arrangements. In: El Zein, F., Suciu, A., Tosun, M., Uludag, A.M., Yuzvinsky, S. (Eds.) Arrangements, Local Systems and Singularities, pp. 177–190. Birkhäuser, Basel (2010)

5. Orlik, P., Terao, H.: Arrangements of Hyperplanes. Springer-Verlag, Berlin (1992)

6. PARI/GP: http://pari.math.u-bordeaux.fr

7. Stanley, R.: Enumerative Combinatorics. Vol. I. Cambridge University Press, Cambridge (1997)